Limiting anthropogenic climate change requires the fast decarbonisation of the electricity system. Renewable electricity generation is determined by the weather and is hence subject to climate change. We simulate the operation of a coarse-scale fully-renewable European electricity system based on downscaled high resolution climate data from EURO-CORDEX. Following a high emission pathway (RCP8.5), we find a robust increase of backup needs in Europe until the end of the 21st century. The absolute increase of the backup needs is almost independent of potential grid expansion, leading to the paradoxical effect that relative impacts of climate change increase in a highly interconnected European system. The increase is rooted in more homogeneous wind conditions over Europe resulting in extensive parallel generation shortfalls. Our results are strengthened by comparison with a large CMIP5 ensemble using an approach based on Circulation Weather Types.